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Centered polygonal number
The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers with a constant number of sides. Each side of a polygonal layer contains one dot more than a side in the previous layer, so starting from the second polygonal layer each layer of a centered k''-gonal number contains ''k more points than the previous layer. Examples Each sequence is a multiple of the triangular numbers plus 1. For example, the centered square numbers are four times the triangular numbers plus 1. In general, the centered k''-gonal numbers are ''k times the triangular numbers plus 1. Thus, the n''th centered ''k-gonal number is kn(n''+1)/2+1 These series consist of the *Centered triangular numbers 1, 4, X, 17, 27, 3X, 54, 71, 91, E4, 11X, 147, 177, 1XX, 224, 261, 2X1, 324, 36X, 3E7, 447, 49X, 534, 591, 631, ... *Centered square numbers 1, 5, 11, 21, 35, 51, 71, 95, 101, 131, 165, 1X1, 221, 265, 2E1, 341, 395, 431, 491, 535, 5X1, 651, 705, 781, 841, ... *Centered pentagonal numbers 1, 6, 14, 27, 43, 64, 8X, E9, 131, 16X, 1E0, 237, 287, 320, 37X, 421, 489, 53X, 5E4, 673, 737, 804, 896, 971, X51, ... *Centered hexagonal numbers 1, 7, 17, 31, 51, 77, X7, 121, 161, 1X7, 237, 291, 331, 397, 447, 501, 581, 647, 717, 7E1, 891, 977, X67, E61, 1061, ... (which are also called hex numbers) *Centered heptagonal numbers 1, 8, 1X, 37, 5E, 8X, 104, 145, 191, 224, 282, 327, 397, 452, 514, 5X1, 675, 754, 83X, 92E, X27, E2X, 1038, 1151, 1271, ... *Centered octagonal numbers 1, 9, 21, 41, 69, X1, 121, 169, 201, 261, 309, 381, 441, 509, 5X1, 681, 769, 861, 961, X69, E81, 10X1, 1209, 1341, 1481, ... (which are exactly the odd squares) *Centered nonagonal numbers 1, X, 24, 47, 77, E4, 13X, 191, 231, 29X, 354, 417, 4X7, 584, 66X, 761, 861, 96X, X84, EX7, 1117, 1254, 139X, 1531, 1691, ... (which include all even perfect numbers except 6) *Centered dekragonal numbers 1, E, 27, 51, 85, 107, 157, 1E5, 261, 317, 39E, 471, 551, 63E, 737, 841, 955, X77, EX7, 1125, 1271, 1407, 156E, 1721, 18X1, ... *Centered elpagonal numbers 1, 10, 2X, 57, 93, 11X, 174, 219, 291, 354, 426, 507, 5E7, 6E6, 804, 921, X49, E84, 110X, 1263, 1407, 157X, 1740, 1911, 1XE1, ... *Centered dozagonal numbers 1, 11, 31, 61, X1, 131, 191, 241, 301, 391, 471, 561, 661, 771, 891, X01, E41, 1091, 1231, 13X1, 1561, 1731, 1911, 1E01, 2101, ... (which are also the star numbers) and so on. The star numbers are exactly the centered dozagonal numbers, thus, the star numbers are exactly the numbers obtained as the concatenation of a triangular number followed by a 1. The triangular numbers are :0, 1, 3, 6, X, 13, 19, 24, 30, 39, 47, 56, 66, 77, 89, X0, E4, 109, 123, 13X, 156, 173, 191, 1E0, 210, ... and the star numbers are :1, 11, 31, 61, X1, 131, 191, 241, 301, 391, 471, 561, 661, 771, 891, X01, E41, 1091, 1231, 13X1, 1561, 1731, 1911, 1E01, 2101, ... Which are exactly the numbers obtained as the concatenation of a triangular number followed by a 1. Thus, if we write a triangular number, and write a digit "1" after this number, then we get a star number. Formula As can be seen in the above diagrams, the ''n''th centered ''k-gonal number can be obtained by placing k'' copies of the (''n−1)th triangular number around a central point; therefore, the n''th centered ''k-gonal number can be mathematically represented by : C_{k,n} =\frac{kn}{2}(n-1)+1. The difference of the n''-th and the (''n+1)-th consecutive centered k''-gonal numbers is ''k(2''n''+1). Just as is the case with regular polygonal numbers, the first centered k''-gonal number is 1. Thus, for any ''k, 1 is both k''-gonal and centered ''k-gonal. The next number to be both k''-gonal and centered ''k-gonal can be found using the formula: : \frac{k^2}{2}(k-1)+1 which tells us that X is both triangular and centered triangular, 21 is both square and centered square, etc. Centered polygonal primes Whereas a prime number p'' cannot be a polygonal number (except the trivial case, i.e. each ''p is the second p''-gonal number), many centered polygonal numbers are primes. In fact, if ''k ≥ 3, k'' ≠ 8, ''k ≠ 9, then there are infinitely many centered k''-gonal numbers which are primes (assuming the Bunyakovsky conjecture). (Since all centered octagonal numbers are also square numbers, and all centered nonagonal numbers are also triangular numbers (and not equal to 3), thus both of them cannot be prime numbers) The centered polygonal primes are *Centered triangular primes 17, 27, 91, 147, 2X1, 3E7, 447, 591, X41, 11X7, 1431, 1877, 2061, 2811, 2931, 40X7, 4531, 50E7, 5761, 6077, 6247, 7E01, 8717, 9371, 9591, X277, E6E7, E947, 11557, 13017, 13E31, 14X87, 15177, 16161, ... *Centered square primes 5, 11, 35, 51, 95, 131, 221, 2E1, 431, 535, 705, 841, 905, 1011, 10E1, 1281, 1465, 1561, 1981, 2111, 2741, 2E51, 42X1, 4E71, 5711, 5905, 6101, 7395, 7X65, 8321, 8565, 8X41, 9591, E005, E561, E835, 101E1, 10781, 11X65, 12171, 13535, 15451, 17865, 19301, 19X71, 1X635, 20201, ... *Centered pentagonal primes 27, 131, 237, 421, 737, 971, 1E17, 2X47, 3321, 34X1, 5367, 6957, 7491, 82E7, X107, X3E7, E0E1, 10467, 15EX1, 17777, 1EX81, 21497, 25961, 26261, ... *Centered hexagonal primes (which are also called hex primes) 7, 17, 31, 51, X7, 1X7, 237, 291, 397, 447, 647, E61, 1061, 1167, 1391, 14E1, 1747, 1X01, 2097, 2537, 26X7, 2EX7, 3177, 3717, 4101, 4307, 4951, 5421, 5E37, 6947, 7001, 7817, 7XE1, 9777, E301, E647, 10EX7, 11721, 12277, 12651, 13217, 13X01, 173X7, 18531, 20961, 22591, 23417, 24801, 25147, 28421, 2X331, 2E307, ... *Centered heptagonal primes 37, 5E, 145, 327, 397, 675, X27, 1151, 1647, 178E, 2181, 2847, 33X5, 4921, 5457, 62X5, 6571, 7207, 8515, X791, EX2E, 14E5E, 153E7, 18077, 19845, 1EX17, 2302E, 254E1, 2748E, 28E15, 29541, 318X5, 334EE, 35861, ... *Centered octagonal primes (not exist, since 8''n(n''+1)/2+1 = (2''n+1)2) *Centered nonagonal primes (not exist, since 9''n''(n''+1)/2+1 = (3''n+1)×(3''n''+2)) *Centered dekragonal primes E, 27, 51, 85, 107, 157, 1E5, 471, 63E, 737, 841, 955, X77, 1125, 1407, 156E, 18X1, 1X6E, 2047, 2837, 3081, 3791, 3X31, 4357, 5585, 5891, 710E, 7E85, 8717, 8XE5, X711, XE3E, E377, E801, 10967, 11E87, 1246E, 12961, 1376E, 14087, 16355, 16901, 17277, 1783E, 1919E, 1X1X1, 1X7E5, 21367, 21X0E, 25987, 27721, 2899E, 29531, 2X83E, 2E3E7, 30755, 31E27, 32725, 33E47, 38171, 3E055, 3E941, 42051, 45161, 46857, 47607, 48385, 4X90E, ... *Centered elpagonal primes 57, 291, 507, 5E7, 921, 1407, 1911, 1XE1, 24X7, 4E71, 6377, 7221, E327, 14891, 23297, 25781, 30947, 31577, 39551, 3X271, 47057, 4X531, 529E7, ... *Centered dozagonal primes (which are also the star primes) 11, 31, 61, 131, 241, 301, 391, 471, 661, 771, 1231, 13X1, 1561, 1911, 1E01, 3291, 3541, 3801, 3X91, 4461, 4761, 5191, 5E91, 6331, 66X1, 7231, 7611, 7X01, 9261, X841, EX71, 10861, 12X91, 139X1, 16911, 17331, 18X31, 19491, 20961, 21E91, 23241, 26431, 27801, 29011, 2E0X1, 2E931, 31E01, 33471, 34161, 35771, 37E41, 3EE31, 42501, 44E61, 459X1, 47691, 4X291, 4E171, 50E61, 52991, 53901, 55791, 5E601, ... and so on. If Bunyakovsky conjecture is true, then centered k''-gonal primes exist (and there are infinitely many such primes) for all ''k except 8 and 9, and the numbers 8 and 9 are in the Catalan's conjecture (i.e. 8 and 9 are the only case of two consecutive perfect powers), besides, the product of 8 and 9 is 60, which is the smallest Achilles number, besides, the concatenation of 8 and 9 is 89, which is the smallest integer such that the factorization of x^n-1 over Q''' includes coefficients other than \pm 1 (i.e. the 89th cyclotomic polynomial, \Phi_{89} , is the first with coefficients other than \pm 1 ), besides, the 3-smooth numbers (or the numbers n'' such that the reciprocal of ''n terminates) ≤ 10 are 1, 2, 3, 4, 6, 8, 9, and 10, all of these numbers except 8 and 9 are divisors of 10 (8 is because it has more prime factors 2 than 10, and 9 is because it has more prime factors 3 than 10) (thus, the numbers of digits of the reciprocal of all these n except 8 and 9 are all 1, while the numbers of digits of the reciprocal of 8 and 9 are 2). Formulae The ''n''th centered ''N''0-gonal number, where ''n'' = 0 gives the central dot, is given by the formula:Where \scriptstyle \,_cP^{(d)}_{N_0}(n)\, is the ''d''-dimensional centered regular convex polytope number with ''N''0 vertices. : \,_cP^{(2)}_{N_0}(n) = N_0\ P^{(2)}_{3}(n) + 1 = N_0\ T_{n} + 1 = N_0 \binom{n+1}{2} + 1 = N_0 + 1,\, where \scriptstyle P^{(2)}_{3}(n) = T_n\, is the ''n''th triangular number. Schläfli-Poincaré (convex) polytope formula Schläfli-Poincaré generalization of the Descartes-Euler (convex) polyhedral formula.Weisstein, Eric W., Polyhedral Formula, From MathWorld--A Wolfram Web Resource. For nondegenerate 2-dimensional regular convex polygons: : {\sum_{i=0}^1 (-1)^i N_i} = N_0-N_1 = V-E = 0,\, where ''N''0 is the number of 0-dimensional elements (vertices ''V'',) ''N''1 is the number of 1-dimensional elements (edges ''E''''') of the convex polygon. Recurrence relation : \,_cP^{(2)}_{N_0}(n) = \,_cP^{(2)}_{N_0}(n-1) + N_0\ n,\, with initial condition : \,_cP^{(2)}_{N_0}(0) = 1.\, Generating function : G_{\{\,_cP^{(2)}_{N_0}(n)\}}(x) = \, Differences : \,_cP^{(2)}_{N_0}(n) - \,_cP^{(2)}_{N_0}(n-1) = N_0\ n = N_0\ P^{(1)}_{1}(n)\, Partial sums : \sum_{n=0}^m {\,_cP^{(2)}_{N_0}(n)} = N_0 \frac{m(m+1)(m+2)}{6} + m = N_0 \binom{m+2}{3} + m = N_0\ P^{(3)}_{4}(m) + m\, Partial sums of reciprocals : \sum_{n=0}^m \frac{1}{\,_cP^{(2)}_{N_0}(n)} = ...\, Sum of reciprocals : \sum_{n=0}^{\infty} \frac{1}{\,_cP^{(2)}_{N_0}(n)} = \frac{2\pi}{N_0 \sqrt{1-\frac{8}{N_0}}} \tan{\bigg( \frac{\pi}{2} \sqrt{1-\frac{8}{N_0}} \bigg)},\ N_0 \neq 8,\, :::::: = \sum_{n=0}^{\infty} \frac{1}{(2n+1)^2} = \frac{\pi^2}{8},\ N_0 = 8.\, Table of formulae and values Category:Pages